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An improved convergence analysis of smoothed aggregation algebraic multigrid. (English) Zbl 1274.65315

Summary: We present an improved analysis of the smoothed aggregation algebraic multigrid method extending the original proof in the work of P. Vaněk et al. [Numer. Math. 88, No. 3, 559–579 (2001; Zbl 0992.65139)] and its modification by P. S. Vassilevski [Multilevel block factorization preconditioners. Matrix-based analysis and algorithms for solving finite element equations. New York, NY: Springer (2008; Zbl 1170.65001)]. The new result imposes fewer restrictions on the aggregates that makes it easier to verify in practice. Also, we extend a result of P. Vaněk [Appl. Math., Praha 57, No. 1, 1–10 (2012; Zbl 1249.65272)] that allows us to use aggressive coarsening at all levels. This is due to the properties of the special polynomial smoother that we use and analyze. In particular, we obtain bounds in the multilevel convergence estimates that are independent of the coarsening ratio. Numerical illustration is also provided.

MSC:

65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs

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References:

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